Tests of Integrability of the Supersymmetric Nonlinear Schrodinger Equation

TL;DR

This paper investigates the integrability of the supersymmetric nonlinear Schrödinger equation using various tests, finding a matrix Lax pair and Painlevé property only for specific parameters, and exploring related structures.

Contribution

It demonstrates the existence of a matrix Lax pair and Painlevé property for certain parameter choices, and examines the Hamiltonian and zero curvature structures in superspace.

Findings

Matrix Lax pair exists for specific parameters

Painlevé property holds only for particular parameter values

A fermionic generalization with Painlevé property is identified

Abstract

We apply various conventional tests of integrability to the supersymmetric nonlinear Schr\"odinger equation. We find that a matrix Lax pair exists and that the system has the Painlev\'e property only for a particular choice of the free parameters of the theory. We also show that the second Hamiltonian structure generalizes to superspace only for these values of the parameters. We are unable to construct a zero curvature formulation of the equations based on OSp(2$|$1). However, this attempt yields a nonsupersymmetric fermionic generalization of the nonlinear Schr\"odinger equation which appears to possess the Painlev\'e property.